C（[—π,π]^n）上多项式逼近的一些结果

首先给出C（[－π，π]n）上连续函数算子序列的一个逼近定理及其在多元多项式逼近方面的一个推论．其次，在所获结果的基础上，再利用高维乘积核构造非正的n维Rogosinski型核及相应的逼近算子，进而又得到了以二阶连续模为逼近阶的高维Rogosinski型逼近定理，     

新二元三角插值多项式的构造

设Ω=[-πxπ,-πyπ],C(Ω)表示关于x,y均以2π为周期的连续函数空间.若f(x,y)∈C(Ω),取结点组为(xk,yl)=(2k+2n 1)π,(2l 2+m 1)πk=0,1,2,…,2n,l=0,1,2,…,2m,则我们获得一个二元三角插值多项式Cn,m(f;x,y)=M1N∑k=2n0∑l=2m0f(xk,yl).1+2∑nα=1cosα(x-xk)+2∑mβ=1cosβ(y-yl)+4∑nα=1∑mβ=1cosα(x-xk)cosβ(y-yl)其中M=2m+1,N=2n+1.为改进其收敛性,本文构造一个新的因子ρα,β,使得带有该因子ρα,β的二元三角插值多项式Ln,m(f;x,y)可以在全平面上一致地收敛到每个连续的f(x,y),且具有最佳逼近阶.     

Jackson算子关于二阶连续模的最佳逼近常数

求了用Jackson算子Jn（f.,x）逼近函数f（x）（∈C2n）时关于二阶连续模的最佳逼近常数：及用阶数不超过n的三角多项式H对连续函数f（x）的最佳逼近En（f）,的上界估计：     

关于Hermite-Fejer插值过程的平均收敛

文［１］讨论了某些非Ｗ－过程的插值算子的加权平均逼近的收敛性和收敛阶．如记Ｈｎ（ｆ；ｘ）为以第二类Ｃｈｅｂｙｓｈｅｖ多项式Ｕｎ（ｘ）的零点作为插值节点，区间［－１，１］上的函数ｆ（ｘ）的Ｈｅｒｍｉｔｅ－Ｆｅｊｅｒ插值算子，［１］中证得：定理Ａ当０＜ｐ...     

二元Lagrange三角插值多项式在Orlicz空间内的逼近

研究了二元函数用一种组合型的三角插值多项式算子逼近的问题.借助连续模这一工具,给出了这类三角插值多项式在Orlicz空间内的逼近定理.     

关于苏联科学院数学研究所在函数逼近论方面的工作（下）

关于苏联科学院数学研究所在函数逼近论方面的工作（下）ＣＡ．捷里亚可夫斯基（原苏联科学院数学研究所）６多元函数逼近多元函数逼近的正逆定理最早是由Ｄ．Ｊａｃｋｓｏｎ［４］和Ｓ．Ｎ．Ｂｅｒｎｓｔｅｉｎ［５］与一元函数的定理同时给出的，对多元函数的系统研究要...     

某些周期卷积类上的最佳单边逼近

设Ｋ满足性质Ｂ（或Ｋ是ＮＣＶＤ核），本文得到了以Ｋ为核的卷积类Ｂ＿ｐ（或Ｋ＿ｐ），１≤ｐ≤∞，在Ｌ＿１尺度下，借助于三角多项式和周期样条的单边逼近的精确值．ｈ，ｆ⊥Ｓ，则ｈ⊥Ｓ，结合（４．７）我们易知Ｆ∈Ｓ￣－（Ｇ￣＊），且Ｆ⊥Ｓ，Ｆ（０）＝０．由引理６得由引理７得进一步地根据定理２，我们可得（４．６）及（４．８）得另一方面．根据［２］（定理１．７．５）及引理８得由（４．８）、（４．１０）定理得证．作者感谢导师孙永生教授的悉心指导．参考文献     

线性流形上矩阵方程AX=B的一类反问题及数值解法

1．引言本文用＊-"m表示全体nX。实矩阵的集合，人表示n阶单位矩阵，汉"m一《ME＊""叫rank（川一r），＊＊"""＝HE＊"""卜"A＝v，＊＊"""一仰E＊"""卜"一M｝，SR；""（SR7""）表示全体7。阶实对称半正定（正定）阵集合．N（A）表示矩阵A的零空间，即N（A）＝（xlAx＝0），ID叫D表示Frobenius范数，A"表示矩阵A的Moors－Penrose广义逆，［EI十表示在Frobenius范数意义下n阶方阵E在SR；""中唯一的最佳k逼近解，即口一［E］＋11-inf。。、。。x，IllE-All．（［E］十求法见文［7］）．还用A三0（A三0）表示A（的k阶顺序主子矩…     

Jacobi多项式零点为结点的Lagrange插值多项式之逼近

对于可微函数ｆ∈Ｃｑ［－１，１］，本文研究以Ｊａｃｏｂｉ多项式Ｊ（α，β）ｎ（ｘ）的零点为结点组之Ｌａｇｒａｎｇｅ插值多项式对ｆ及其导数的同时逼近，证明不等式Ｌ（ｓ）ｎ（ｆ，α，β，ｘ）－ｆ（ｓ）（ｘ）＝Ｏ（１）Δ－ｓｎ（ｘ）Δｑｎ（ｘ）ω（ｆ（ｑ），Δｎ（ｘ））ｌｏｇｎ｛＋（１－ｘ＋ｎ－１）－α－１２ｎ－ｑω（ｆ（ｑ），ｎ－１）｝，在［０，１］上对于ｓ＝０，１，２，…，ｑ一致成立，其中Δｎ（ｘ）＝ｎ－１１－ｘ２＋ｎ－２     

关于《L_p范数下凸约束最佳逼近的特征》中一个引理的注记

最近．许树声［１］撰文给出了对Ｌ_ｐ范数下凸约束最佳逼近的特征定理．并研究了该定理的苦干应用．然而，文［１］引理１证明中部分地方有误．本文给予了纠正，予以重新证明     

On Some Semigroups on the Complex Plane

It is shown that the sets $C(\alpha)=\left\{z\in\dC: |z\sin\alpha\pm i\cos \alpha|\le 1\right\}$, where $\alpha\in (0,\pi/2)$ form multiplicative semigroups on the complex plane $\dC$. We prove that the semigroups $C(\alpha)$ and $C(\beta)$ are not isomorphic when $\alpha\ne \beta$ and the unique automorphisms of the semigroup $C(\alpha)$ are the mappings $\Phi(z)=z$ and $\Phi(z)=\overline z$. All continuous semicharacters of the semigroups $C(\alpha)$ and all continuous automorphisms of the closed unit disk are described. Other examples of semigroups on the complex are obtained by transformations of $C(\alpha).$     

Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles . The same condition is necessary and sufficient for the triple to be interior angles of a spherical triangular membrane.

     

On Estimate of Complete Trigonometric Suns

In this paper, the following theorem is proved: Let f(x)=a_kx^k+\cdots+a_1x+a_0 be a polynomial with integral coefficients such that (\alpha_1,\cdots,\alpha_k,q)=1, where q is a positive integer. Then, for k\geq 3, $[|\sum\limits_{x = 1}^q {{e^{2\pi if(x)/q}}} | \le {e^{2k}}{q^{1 - 1/k}}\]$     

ON THE LEAST DOMINANT CONTINUOUS MODULUS AND ITS APPLICATION

o.IntroductionInthispaperwedealwithqualltitativeKorovkintypetheoremsfortheaPproalmationbyboundedlinearoperatorsdefinedonC(X),andinparticularbypositiveones.HereC(X)=CR(X,d)denotestheBanachlatticeofreal-valuedcontinuousfunctionsdefinedonthecompactmetricspace(X,d)withnormgivenbylIfIlx=ma-xlf(x)I,xEX.WealsoassumethatXhasdiameterd(X)>o.ThefirstsuchtheoremforgeneralpositivelinearoperatorsandX=[a,b]equippedwiththeeuclidiandistanceisduetoR.Mamedovl4].Forspaces(X,d)beingmetricallyconvexinthes…     

SOME EXTENSIONS OF PALEY-WIENNER THEOREM

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭｏｔｉｖａｔｉｏｎＴｈｅｃｌａｓｉｃａｌＳｈａｎｎｏｎ’ｓｓａｍｐｌｉｎｇｔｈｅｏｒｅｍｈｏｌｄｓｄｕｅｔｏｔｈｅｆｏｌｏｗｉｎｇｔｗｏｒｅａｓｏｎｓ：（ｉ）ｑ（ｔ，ｕ）＝ｓｉｎπ（ｔ－ｕ）π（ｔ－ｕ）ｉｓｔｈｅ...     

Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space $$L_2 $$

To a function $f \in L_2 [ - \pi ,\pi ]$ and a compact set $Q \subset [ - \pi ,\pi ]$ we assign the supremum $\omega (f,Q) = \sup _{t \in Q} ||f( \cdot + t) - f( \cdot )||_{L_2 [ - \pi ,\pi ]} $ , which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function f by trigonometric polynomials of degree $n - 1$ in the space $L_2 [ - \pi ,\pi ]$ and the modulus of continuity $\omega (f,Q)$ . It follows from results due to Chernykh that $K(n,Q) \geqslant 1/\sqrt 2 $ and $K(n,[0,\pi /\pi ]) = 1/\sqrt 2 $ . On the strength of a result of Yudin, we show that if the measure of the set Q is less than $\pi /n$ , then $K(n,Q) >1/\sqrt 2 $ .     

Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.     

ON THE L~p-BOUNDEDNESS OF SEVERAL CLASSES OF PSEUDO-DIF FERENTIAL OPERATORS

To indicate precisely the requirements for smoothness of symbols,generalizations ofH(?)rmander's classes of symbols,S_(ρ,k(?),v)~m and S_(ρ,k(?),v(?))~m,are introduced.The main results areas follows:(1)An optimal L~2-boundedness result is obtained for the pseudo-differentialoperators with double symbols(amplitude)a(x,(?),y);(2)By means of the interpolationtheorem due to Fefferman and Stein,new L~p-boundedness results are established.Theseresults are not only sharp with respect to upper index,but also sharp(p≥2)or almost sharp(1相似文献   

有循环极大子群的素数幂阶群的作用是边传递的图(II)

假定Γ是一个有限的、单的、无向的且无孤立点的图,G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.结果为:设图Γ含有一个阶为pn(p是素数,n≥2)的自同构群,且G有一个极大子群循环,则Γ是G-边传递的,当且仅当Γ同构于下列图之一1)pmK1,pn-1-m,0≤m≤n-1;2)pmK1,pn-m,0≤m≤n;3)pmKp,pn-m-1,0≤m≤n-2;4)pn-mCpm,pm≥3,m＜n;5)2n-2K1,1;6)pn-1-mCpm,pm≥3,m≤n-1;7)2pn-mCpm,pm≥3,m≤n-1;8)2pn-mK1,pm,0≤m≤n;9)pn-mK1,2pm,0≤m≤n;10)pn-mK2,pm,0＜m≤n;11)C(2pn-m,1,pm);12)pkC(2pm-k,1,pn-m),0＜k＜m,0＜m≤n;13)(t-s,2m)C(2m 1/(t-s,2m),1,2n-1-m),其中0≤m≤n-1,2n-2(s-1)≡0(mod 2m),t≡1(mod 2),s(≠)t(mod 2m),1≤s≤2m,1≤t≤2n-1;14)∪p i=1 Ci p n-1,其中Ci p n-1=Ca1a1 [1 (i-1)pn-2]a 1 2[1 (i--1)p n-2]…a 1 (pn-1-1)[1 (i-1)p n-2]≌Cp n-1,i=1,2,…,p;15)∪2 i=1 Ci 2n-1,其中Ci 2n-1=Ca1a 1 [1 (i-1)(2n-2-1)]a1 2[1 (i-1)(2n-2-1)]…a1 (2n-1-1)[1 (i-1)(2n-2-1)]≌C2n-1,i=1,2.